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If a second-order differential equation has a characteristic equation with complex conjugate roots of the form r 1 = a + bi and r 2 = a − bi, then the general solution is accordingly y(x) = c 1 e (a + bi )x + c 2 e (a − bi )x. By Euler's formula, which states that e iθ = cos θ + i sin θ, this solution can be rewritten as follows:
Characteristic (algebra) – Smallest integer n for which n equals 0 in a ring; Characteristic (exponent notation) – Mathematical function; Classification theorem – Describes the objects of a given type, up to some equivalence; Euler characteristic – Topological invariant in mathematics
The Ritt–Wu process, first devised by Ritt, subsequently modified by Wu, computes not a Ritt characteristic but an extended one, called Wu characteristic set or ascending chain. A non-empty subset T of the ideal F generated by F is a Wu characteristic set of F if one of the following condition holds T = {a} with a being a nonzero constant,
In mathematics, the method of characteristics is a technique for solving partial differential equations.Typically, it applies to first-order equations, though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equation.
The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor (possibly negative). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. Its eigenvectors are those ...
The characteristic of a ring R is the natural number n such that n is the kernel of the unique ring homomorphism from to R. [a]The characteristic is the natural number n such that R contains a subring isomorphic to the factor ring /, which is the image of the above homomorphism.
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
In mathematics, a variable (from Latin variabilis, "changeable") is a symbol, typically a letter, that refers to an unspecified mathematical object. [1] [2] [3] One says colloquially that the variable represents or denotes the object, and that any valid candidate for the object is the value of the variable.